BENFORD ’S LAW

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BENFORDS LAW


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BENFORDS LAW

History

What is Benfords Law

Types of Data That ConformUses in Fraud Investigations

Examples

Other uses of Benfords LawCautions When Using Benfords Law


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HISTORY

Simon Newcomb 1881

Frank Benford 1938

Roger Pinkham 1961 Theodore Hill 1995


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History Accounting Data

Benfords Law was first used by

accountants in late 1980

Nigrini,


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WHAT IS BENFORDS LAW

BENFORDS LAW FORMULA

The probability of any number d from1 through 9 being the first digit is.

Log10 (1 + 1/d)


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WHAT IS BENFORDS LAW?

Benfords law gives the probability of

obtaining digits 1 through 9 in eachposition of a number.

For example, 3879

3 - first digit

8 - second digit

7 - third digit

9 fourth digit


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WHAT IS BENFORDS LAW

Most people assume the probability is 1/9that the first digit will be 1 - 9

This would mean digits are equally likely tooccur, but this is not the case

According to Benfords Law the probability

of obtaining a 1 in the first digit position is30.1%


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Expected Frequencies Based on

Benfords Law

Digit 1st

Place 2nd

Place 3rd

Place 4th

Place

0 0.11968 0.10178 0.10018

1 0.30103 0.11389 0.10138 0.10014

2 0.17609 0.19882 0.10097 0.1001

3 0.12494 0.10433 0.10057 0.10006

4 0.09691 0.10031 0.10018 0.10002

5 0.07918 0.09668 0.09979 0.09998

6 0.06695 0.09337 0.0994 0.09994

7 0.05799 0.0935 0.09902 0.0999

8 0.05115 0.08757 0.09864 0.09986

9 0.04576 0.085 0.09827 0.09982

Source: Nigrini, 1996.


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Benfords Law


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http://testingbenfordslaw.com/rails-core-line-count


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Logic Behind Benfords Law

If a data entry begins with the digit 1 it hasto double in size (100%)before it beginswith the digit 2

If a data entry begins with the digit 9 it onlyhas to be increased by 11% in order forthe first digit to be a 1


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Types of Data That ConformWhen Benford Analysis Is

Likely Used

Examples

Sets of numbers that result from

mathematical combination of

numbersResult comes

from two distributions

Accounts receivable (number sold * price),

Accounts payable (number bought * price)

Transaction-level dataNo

need to sampleDisbursements, sales, expenses

On large data setsThe more

observations, the betterFull years transactions

Accounts that appear to conformWhen the mean of a set of

numbers is greater than the

median and the skewness is

positive

Source: Durtschi, 2004, 24.

Most sets of accounting numbers


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Types of Data That Do NotConform

When Benford Analysis IsNot

Likely UsedExamples

Data set is comprised of assigned

numbers

Check numbers, invoice numbers, zip

codes

Numbers that are influenced byhuman thought

Prices set at psychological thresholds($1.99, ATM withdrawals

Accounts with a large number of

firm-specific numbers

An account specifically set up to record

$100 refunds

Accounts with a built in

minimum or maximum

Set of assets that must meet a threshold to

be recorded

Where no transaction is recorded

Source: Durtschi, 2004, 24.

Thefts, kickbacks, contract rigging


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Types of Data That Conform

Accounts payable data

Accounts receivable data

Estimations in the general ledger

Relative size of inventory unit pricesamong locations

New combinations of selling prices

Customer refunds

Duplicate payments


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Uses in Fraud Investigations

Invented or altered numbers are not likelyto follow Benfords Law

Human choices are not random

1993, State of Arizona v.Wayne JamesNelson


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Examples of Benfords Law

Benfords Law was used to analyze the

first 2 digits of accounts payable data for aNASDAQ-listed software company


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Examples of Benfords Law

Bank audit


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Small Business Owner Example

Small business owner expanded his one-storefamily-owned business into a four-store chain

Had to relinquish some hands-on control with

the expansion Concerned about bookkeeping errors or

possibility of fraud

Owner used Excel program based on BenfordsLaw to analyze the stores disbursement data


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Small Business Owner Example(Source: Rose, 2003)

First Digits Test

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9

Digit

Rate Benford

Sample


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Second Digits Test

00.05

0.1

0.15

0.2

0.250.3

0.35

0 1 2 3 4 5 6 7 8 9

Digit

Rate Benford

Sample


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Small Business Owner Example First 2 Digits Test

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10

14

18

22

26

30

34

38

42

46

50

54

58

62

66

70

74

78

82

86

90

94

98

Two Digit Pair

Rate Benford

Sample


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Benfords Law Analysis cont.

First Two-Digits Test 56 & 67 are the two digit combinations that

appear more frequently than expectedOwner pulled sample of disbursements

starting with the 56 & 67 sequences

Discovered pymts to unfamiliar vendor Addtl invest revealed vendor did not exist

pymts going to personal acct


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Other Uses of Benfords Law

Year 2000 problem

allocating computer disk space

detect irregularities in clinical trials


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Other Uses of Benfords Law

Demographic models of populationstatistics & vital statistics

Customs officials

ballot fraud in Ukraines republic elections


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Cautions

Not necessarily fraud

False positives

Certain types of fraud will not be detectedusing Benfords Law analysis


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Conclusion

Useful tool

Digital analysis


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Suggested Reading

Works Cited Amazing Applications of Probability and Statistics,

http://www.intuitor.com/statistics/Benfords%20Law.html, accessed 9/21/2004. Browne, Malcolm W. Following Benfords Law, or Looking Out for No. 1,

http://www.rexswain.com/benford.html, accessed 9/21/2004 (From The New York Times,Tuesday, August 4th, 1998).

Durtschi, Cindy and William Hillison and Carl Pachini. The Effective Use of Benfords Lawto Assist in Detecting Fraud in Accounting Data, Journal of Forensic Accounting1524-

5586/Vol.V(2004): 17-34. Findings and Observations, http://www.nigrini.com/images/ForensicAccountantReport.html,

accessed 9/22/2004. Matthews, Robert. Benford Bend, http://www.euromoneyplc.com, accessed 12/1/2004 (From

World Link; May/June 2000, 10-12). Nigrini, Mark. Taxpayer Compliance Application of Benfords Law, Journal of the American

Taxation Association. 18(1), 1996: 72-92. Nigrini, Mark. Ive Got Your Number, http://www.aicpa.org/pubs/jofa/may1999/nigrini.html,

accessed 9/22/04 (From Journal of Accountancy, May 1999). Rose, Anna and Jacob Rose. Turn Excel Into a Financial Sleuth,

http://web9.epnet.com.ezproxy.udayton.edu, accessed 12/1/2004 (From Journal of Accountancy,August 2003, Vol 196 Issue 2, 58-61).

Walthoe, Jon and Robert Hunt and Mike Pearson. Looking Out For Number One,http://pass.maths.org.uk/issue9/features/benford, accessed 11/24/2004 (From MillenniumMathematics Project, University of Cambridge, Sept. 1999).
http://www.intuitor.com/statistics/Benford's%20Law.htmlhttp://www.rexswain.com/benford.htmlhttp://www.nigrini.com/images/ForensicAccountantReport.htmlhttp://www.euromoneyplc.com/http://www.aicpa.org/pubs/jofa/may1999/nigrini.htmlhttp://web9.epnet.com.ezproxy.udayton.edu/http://web9.epnet.com.ezproxy.udayton.edu/http://pass.maths.org.uk/issue9/features/benfordhttp://pass.maths.org.uk/issue9/features/benfordhttp://web9.epnet.com.ezproxy.udayton.edu/http://www.aicpa.org/pubs/jofa/may1999/nigrini.htmlhttp://www.euromoneyplc.com/http://www.nigrini.com/images/ForensicAccountantReport.htmlhttp://www.rexswain.com/benford.htmlhttp://www.intuitor.com/statistics/Benford's%20Law.html

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